Science & Technology Q&A for Kids (and others) [April 10, 2026]

First question from Girdles is, is there an actual use case for AI math proofs? Would you, me, ever actually find them useful in my work? Well, let's see.

Let's talk a little bit about proofs.

Then we'll talk about proofs in math.

Then we'll talk a bit about AI, AI doing proofs, and so on.

And then we'll talk about places where I at least might find them useful.

So what's the idea of a proof? I think sort of the notion of proofs kind of originated with old Euclid back in maybe 2000, a bit more than 2000 years ago, where people were saying, you know, I have this geometrical, I have something where I'm drawing a circle, I draw another circle that intersects with it, and I draw lines from here to there.

And it's like, is that line, is that angle between two lines a right angle, for example? And you could draw a picture of the circles intersecting and so on.

You could kind of measure, yeah, it seems like it's more or less a right angle.

But the question was, can you actually prove that it's a right angle? Can you give essentially a logical, a structured argument that says it must be a right angle, not just it seems to be roughly a right angle? And how do you make a proof? Well, you have to start with certain assumptions, certain axioms.

So for example, in the geometry of Euclid, a famous axiom was that two parallel lines never cross.

That axiom is a reasonable thing to assume if you're doing geometry in a plane.

It turns out not to be true about physical space.

The discovery 100 years ago that space is curved.

It's curved by the presence of mass and so on.

That's what leads to gravity.

That curvature in space makes Euclid's axiom that two parallel lines never cross not be true for physical space.

You can still, if you're just doing theoretical math, you can still say, we're just going to take it as an assumption, as an axiom, the two parallel lines will love across.

So in Euclidean geometry, for example, there are about 10 axioms like that.

Like there are axioms about if there's, if, well, for example, another famous one is if A is equal to B and B is equal to C, then A is equal to C.

Now, that will be true for some meanings of equal, but for example, if you substitute for equal is a friend of, if A is a friend of B, B is a friend of C, in real life, it absolutely does not follow that A is a friend of C.

But it is assumed for the notion of things being equal in a mathematical sense that that so-called transitivity is true.

So in any case, you start in making a proof from a set of axioms, a set of abstract assumptions about things.

Those assumptions may or may not correspond to what is true in the real world if you identify these parallel lines with actual lines in space and things like this.

But it's a set of abstract assumptions that one makes.

And then you have to say, well, what can we prove from those abstract assumptions? Take this assumption and that one, this axiom, that axiom, you combine them together, you get another thing and so on.